If $\omega_+(x)\cap\omega_-(x)$ contains a regular point, then $x$ is periodic

Question

Suppose we have the dynamical system coming from a two-dimensional autonomous ODE $\dot{x}=f(x)$. Suppose $\omega_+(x)\cap\omega_-(x)$ contains a regular point (i.e. a non-fixed point) where $\omega_{\pm}(x)$ denotes the forward/backward $\omega$-limit set of $x$. The claim is then that $x$ must be a (regular) periodic point. This is problem 7.13 in Teschl's Ordinary Differential Equations and Dynamical Systems. I am not sure how one would proceed with showing this. Is there maybe some way this follows from Poincare-Bendixson?